Translations
Define a translation as a rigid motion that shifts every point by the same vector. Perform translations on the coordinate plane using coordinate rules (x, y) → (x+a, y+b) and recognize that translations preserve size and shape.
A lesson of Geometry, in Transformations, about 10 minutes.
By the end of this lesson, you will be able to translate figures on the coordinate plane using vector notation, apply the translation rule $(x, y) \to (x + a, y + b)$, and verify that translations are isometries (preserve distance and angle).
The Big Idea
A translation slides every point of a figure the same distance in the same direction. It's the simplest rigid motion: no turning, no flipping, just sliding. The pre-image and image are congruent because every distance and every angle is preserved exactly.
Definition
A translation by vector $\langle a, b \rangle$ maps every point $(x, y)$ to $(x + a, y + b)$.
The vector $\langle a, b \rangle$ specifies:
- $a$: horizontal shift (positive = right, negative = left)
- $b$: vertical shift (positive = up, negative = down)
Notation:
- Pre-image: $\triangle ABC$
- Image: $\triangle A'B'C'$ (read "A prime, B prime, C prime")
Properties of Translations
- Isometry: preserves distance — $AB = A'B'$
- Preserves angle measure: $\angle ABC = \angle A'B'C'$
- Preserves orientation: no flipping — same order of vertices
- Maps parallel lines to parallel lines
- Every point moves the same amount: $\overrightarrow{AA'} = \overrightarrow{BB'} = \overrightarrow{CC'}$
Example 1
Translate $\triangle ABC$ with $A(1, 2)$, $B(4, 2)$, $C(3, 5)$ by vector $\langle 3, -4 \rangle$.
$$A' = (1+3, 2-4) = (4, -2)$$ $$B' = (4+3, 2-4) = (7, -2)$$ $$C' = (3+3, 5-4) = (6, 1)$$
Example 2 — Finding the Translation Vector
$P(2, 5)$ maps to $P'(6, 1)$. Find the translation vector.
$$\langle 6-2, 1-5 \rangle = \langle 4, -4 \rangle$$
Example 3 — Algebraic Rule
What translation maps $A(x, y)$ to $A'(x - 3, y + 7)$?
Translation vector: $\langle -3, 7 \rangle$ — shift 3 left and 7 up.
Translations and Vectors
A translation is fully described by a vector. Two translations with the same vector move every figure identically. The composition of two translations (translating by $\langle a, b \rangle$ then by $\langle c, d \rangle$) equals a single translation by $\langle a+c, b+d \rangle$.
Common Mistakes
Mistake 1: Confusing direction of $a$ and $b$
$a$ is horizontal ($x$), $b$ is vertical ($y$). A vector $\langle -5, 3 \rangle$ shifts left 5 and up 3.